(Sums of normal random variables) Let X be independent random variables where XN N(2,5) and Y...
1. Let X1, X2, , Xn be independent Normal μ, σ2) random variables. Let y,-n Σ_lx, denote a sequence of random variables (a) Find E(y,) and Var(y,) for all n in terms of μ and σ2. (b) Find the PDF for Yn for alln. (c) Find the MGF for Yn for all n.
8.7-11. Let Y1,Y2, ...,Yn be n independent random variables with normal distributions N(Bx;,02), where X],x2,...,xn are known and not all equal and B and 2 are unknown parameters (a) Find the likelihood ratio test for Ho: B = 0 against H: B+0. (b) Can this test be based on a statistic with a well-known distribution?
Let X1, X2, , xn are independent random variables where E(X)-? and Var(X) ?2 for all i = 1, 2, , n. Let X-24-xitx2+--+Xy variables. is the average of those random Find E(X) and Var(X).
Let Xi, X2, , xn be independent Normal(μ, σ*) random variables. Let Yn = n Ση1Xi denote a sequence of random variables (a) Find E(%) and Var(%) for all n in terms of μ and σ2. (b) Find the PDF for Yn for all n c) Find the MGF for Y for all n
Let X and Y be independent normal random variables with parameters E[X] =ux, E[Y] = uy and Var(X) = x, Var(Y) = Oy. Indicate whether each of the following statements is true or false. Notation: fx,y (x, y), fx(x), fy (v) denote the joint and marginal PDFs of X and Y , respectively; $(x) is the CDF of a standard normal random variable with zero mean and unit variance. E[XY]=0
9. Let X and Y be two random variables. Suppose that σ = 4, and σ -9. If we know that the two random variables Z-2X?Y and W = X + Y are independent, find Cov(X, Y) and ρ(X,Y). 10. Let X and Y be bivariate normal random variables with parameters μェー0, σ, 1,Hy- 1, ơv = 2, and ρ = _ .5. Find P(X + 2Y < 3) . Find Cov(X-Y, X + 2Y) 11. Let X and Y...
1. Let X1, ..., Xn, Y1, ..., Yn be mutually independent random variables, and Z = + Li-i XiYi. Suppose for each i E {1,...,n}, X; ~ Bernoulli(p), Y; ~ Binomial(n,p). What is Var[Z]?
Let the independent normal random variables Y1,Y2, . . . ,Yn have the respective distributions N(μ, γ 2x2i ), i = 1, 2, . . . , n, where x1, x2, . . . , xn are known but not all the same and no one of which is equal to zero. Find the maximum likelihood estimators for μ and γ 2.
1) Let X and Y be random variables. Show that Cov( X + Y, X-Y) Var(X)--Var(Y) without appealing to the general formulas for the covariance of the linear combinations of sets of random variables; use the basic identity Cov(Z1,22)-E[Z1Z2]- E[Z1 E[Z2, valid for any two random variables, and the properties of the expected value 2) Let X be the normal random variable with zero mean and standard deviation Let ?(t) be the distribution function of the standard normal random variable....
Practice problems using various statistical methods If n independent random variables X have normal distributions with means μ and the standard deviations σ , then determine the distribution of a. I. X-E(X) var(X) C. 2. If n independent random variables Xi have normal distributions with means μί and the standard deviations σί, then determine the distribution of a. b. Y -a1X1 + a2X2+ + anXn (ai constant) X-E(X) Vvar(X) 3. What is CLT? Proof briefly? What are t-, Chi-squared- and...